I was reviewing an article about the effectiveness of a vaccine and it expressed these in % effectiveness = 1 - odds ratio. So far, so good.

But they showed confidence intervals around the %'s, both in text (e.g. 72%, 95% CI = 33.9% to 88.2%) and in a graph of those numbers (they had a bunch of results to display).

It struck me that the CIs are asymmetric (i.e 72-34 = 38, while 88-72 = 16) and this seemed a bit odd, especially for the graph. Should such graphs use log odds which has symmetric CIs?

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    $\begingroup$ Why are asymmetrical confidence intervals odd? $\endgroup$ Sep 25, 2018 at 15:27
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    $\begingroup$ A cute way to visualize CIs for proportions (that suffer from a similar problem -- you want them to be within [0,1] rather than fly around as the asymptotic normality carelessly thinks) is inchworm plots github.com/BiostatGlobalConsulting/inchworm-plots-stata. They show a density function of the approximating distribution which is probably a Beta. $\endgroup$
    – StasK
    Sep 25, 2018 at 15:51
  • $\begingroup$ Why not simply ask the authors to explain/substantiate the CIs they used? $\endgroup$
    – Jim
    Oct 2, 2018 at 12:05
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    $\begingroup$ More interesting than the CI (which transforms perfectly) and possible asymmetry of that interval (relative to the mean) is the question about the mean (which does not transform correctly). When one computes mean and CI on some scale but presents them on another scale, then the mean becomes ambiguous (is it $E(f(X))$ or is it $f(E(X))$). $\endgroup$ Oct 2, 2018 at 14:53

2 Answers 2


A pretty standard approach (most common for relative effect measures such as odds ratios, risk ratios, rate ratios and hazard ratios) is to use a figure with axes that show probabilities (i.e. all the axis labels show probabilities), but which are on the logit scale. I.e. the distance from 0.5 to 0.73 (0 to 1 on the logit scale) is the same as from 0.73 to 0.88 (1 to 2 on the logit scale). This is particularly useful, when multiple probabilities are being shown that people might visually compare to each other on the graph.

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    $\begingroup$ I think I understand what you mean, but if it's probabilities then how can there be any distance from 1 to 2? $\endgroup$
    – amoeba
    Oct 2, 2018 at 13:26
  • $\begingroup$ Sorry, was thinking more of odds ratios. Will update with probability example. $\endgroup$
    – Björn
    Oct 2, 2018 at 13:40

I would say that it is beneficial to show the asymmetric confidence intervals in order for readers to (better) realize that the odds ratio is asymmetric around its null value. Especially showing the figure rather than reporting the 95% CIs in the text makes this stronger/clearer.

The same of course holds for other types of ratios often used, e.g., risk ratio, or hazard ratios.

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    $\begingroup$ +1, I don't see any problem with showing asymmetric CIs either. $\endgroup$
    – amoeba
    Oct 2, 2018 at 13:27

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